## Advanced Boundary Value Problems I

Fall 2004, MWF 1300-1350 V 128

Dr. Thomas Hillen
Associate Professor

phone: 492-3395,

e-mail: thillen@ualberta.ca

ww.math.ualberta.ca/~thillen

University of Alberta
Department of Mathematical and Statistical Sciences

## Assignments: Due at 3:45 PM on due date:

No late assignments will be accepted!

 No Due Date Exercises Solutions 1 Fri, Sep 24, 04 Assignment #1 Solutions #1 2 Fri, Oct 08, 04 Assignment #2 Solutions #2 3 Extended to: Fri, Oct 29, 04 Assignment #3 Solutions #3 4 Fri, Nov 19, 04 Assignment #4 Solutions #4 5 Fri, Dec 06, 04 Assignment #5 Solutions #5 6 No 6th Assignment

### Text:

N.H. Asmar,  Partial Differential Equations with Fourier Series and Boundary Value Problems, Prentice Hall, New Jersey, 2nd ed, 2005.

Midterm Exam:

Wednesday, Oct 27, 2004, 1-1:50 PM in V 128.

Final Exam:

Wednesday, Dec 15, 2004, 2 PM in Univ Pavillon

## Syllabus and Course Notes:

 1. A Preview of Applications and Techniques {1.1} What is a Partial Differential Equation,  Method of Characteristics and Examples The Advection Equation with source term D'Almeberts solution of the wave equation {1.2} Solving and Interpreting a Partial Differential Equation (1.1) Examples 1 and 5, Maple file 2. Fourier Series {2.1} Periodic Functions {2.2} Fourier Series,{2.3} Fourier Series of Functions with Arbitrary Periods (part 1), (part 2), (part 3) {2.4} Half-Range Expansions: The Cosine and Sine Series {2.6} Complex Form of Fourier Series {2.7} Forced Oscillations (2.2) Example 1,  (2.2) Example 2,  Example 3 (2.4) Example 3. Partial Differential Equations in Rectangular Coordinates {3.1} Partial Differential Equations in Physics and Engineering, Classification of Second-Order Linear Equations {3.2} Modeling: Vibrating Strings and the Wave Equation {3.3} Solution of the One Dimensional Wave Equation:  The Method of Separation of Variables {3.4} D'Alembert's Method {3.5} The One Dimensional Heat Equation {3.6} Heat Conduction in Bars: Varying the Boundary Conditions (3.6) Example 2 {3.7} The Two Dimensional Wave and Heat Equations (corrected version) {3.8} Laplace's Equation in Rectangular Coordinates    Summary              Example (3.3) Example 1 (3.3) Example 2 (3.4) Example 2 (3.5) Example 1 4. Partial Differential Equations in Polar and Cylindrical Coordinates {4.1} The Laplacian in Various Coordinate Systems 6. Sturm-Liouville Theroy with Engineering Applications {6.1} Orthogonal Functions {6.2} Sturm-Liouville Theory      Part 1          Part 2 {4.2} Vibrations of a Circular Membrane: Symmetric Case {6.3} The Hanging Chain 7. The Fourier Transform and its Applications {7.1} The Fourier Integral Representation {7.2} The Fourier Transform {7.3} The Fourier Transform Method {7.4} The Heat Equation and Gauss's Kernel {7.6} The Fourier Cosine and Sine Transforms {7.7} Problems Involving Semi-Infinite Interval Not relevant for final exam: {7.8} Generalized Functions

# Outline

### Lecture A1:

M W F 1:00 - 1:50 V 128
Thomas Hillen, 575 CAB

office hours: W 4:00 - 5:00, R 3:00 - 4:00, or by appointment
telephone: 492-3395
e-mail: thillen\@ualberta.ca
web: http://www.math.ualberta.ca/~thillen/math300/math300.html

### Lecture A2:

M W F 1:00 - 1:50 P 146
Ed Leonard, 679 CAB
office hours: M W F 2:00 - 3:00, or by appointment
telephone: 492-2388
e-mail: isaac@cs.ualberta.ca
web: http://www.cs.ualberta.ca/~isaac/

 Homework 20% Midterm Exam: Wednesday, Oct 27, 2004, 1-1:50 PM in V 128. 30% Final Exam: 50%

Deferred Exam: January 15, from 9:00 until 12:00, in CAB 243.
No calculators are allowed during the exams.

There will be no marks for class participation or in class presentations.

The final grades are not curved. The minimum passing grade (D) corresponds to 50%.
The grades of C-, C, C+ correspond roughly to 63 - 73%, the grades of B-, B, B+ correspond
roughly to 74 - 88%, and the grades of A-, A, A+ correspond roughly to 89 - 100%. I reserve the right to adjust the scale upwards (as to give better grades).

### Assignments:

There will be 6 assignments given during the term, one every two weeks. Each assignment will consist of 10 problems of equal weight taken from the text.Sections A1 and A2 have the same assignments.

Assignments are to be submitted in the appropriate Section Boxes on the 3rd floor in CAB, before 3:45 p.m. on the due date.

No late assignments will be accepted.

The first page of your assignment should contain only your  Name and  Lecture Section.

### Calendar Description:

MATH 300 Advanced Boundary Value Problems I:
Derivation of the classical partial differential equations of applied mathematics, solutions using separation of variables. Fourier expansions and
their application to boundary value problems. Introduction to the Fourier transform. Emphasis on building an appropriate mathematical model from a physical problem, solving the mathematical problem, and carefully interpreting the mathematical results in the context of the original physical problem.
Prerequisiteis: Math 201 and 209 or equivalents. Notes: (1) Open only to students in Engineering, Specialization Computing Science, Specialization Physics, and Specialization Geophysics. (2) This course may not be taken for credit if credit has already been obtained in MATH 337.

### Policies:

Policy about course outlines can be found in Section 23.4(2) of the University Calendar.

The University of Alberta is committed to the highest standards of academic integrity and honesty. Students are expected to be familiar with these standards regarding academic honesty and to uphold the policies of the University in this respect. Students are particularly urged to familiarize themselves with the provisions of the Code of Student Behavior (online at www.ualberta.ca/secretariat/appeals.htm) and avoid any behavior which could potentially result in suspicions of cheating, plagiarism, misinterpretation of facts and/or participation in an offence. Academic dishonesty is a serious offence and can result in suspension or expulsion from the University.